Expanders and right-angled Artin groups

We define the notion of a vector space expander, which is a sequence of vector spaces equipped with vector space valued bilinear pairings, and which satisfy a suitably defined Cheeger constant bound. These objects generalize graph expanders in the following sense: when the vector spaces come from the first cohomology of a right-angled Artin groups with coefficients in an arbitrary field and the bilinear pairings are the cup product on the cohomology ring, a sequence of cohomology vector spaces forms a vector space expander family if and only if the sequence of graphs defining the corresponding right-angled Artin groups forms a graph expander family. We thus obtain an intrinsic characterization of expander graphs via right-angled Artin groups.